An insightful, concise, and rigorous treatment of the basic theory of convex sets and functions in finite dimensions, and the analytical/geometrical foundations of convex optimization and duality theory.

Convexity theory is first developed in a simple accessible manner, using easily visualized proofs. Then the focus shifts to a transparent geometrical line of analysis to develop the fundamental duality between descriptions of convex sets and functions in terms of points and in terms of hyperplanes. Finally, convexity theory and abstract duality are applied to problems of constrained optimization, Fenchel and conic duality, and game theory to develop the sharpest possible duality results within a highly visual geometric framework.

The book is supplemented by a long web-based chapter (over 150 pages), which covers the most popular convex optimization algorithms (and some new ones), and can be downloaded from this page.

The book may be used as a text for a theoretical convex optimization course; the author has taught several variants of such a course at MIT and elsewhere over the last ten years. It may also be used as a supplementary source for nonlinear programming classes, and as a theoretical foundation for classes focused on convex optimization models (rather than theory).
It is an excellent
supplement to several of our books: Nonlinear
Programming (Athena Scientific, 1999), Network Optimization
(Athena Scientific, 1998), Introduction to Linear
Optimization (Athena Scientific, 1997), and Network Flows and Monotropic Optimization (Athena Scientific, 1998).

From the review by Panos Pardalos (Optimization Methods and Sofware, 2010):(Full Review)

"The textbook, Convex Optimization Theory (Athena) by Dimitri Bertsekas, provides a concise,
well-organized, and rigorous development of convex analysis and convex optimization theory.
Several texts have appeared recently on these subjects ... The text by
Bertsekas is by far the most geometrically oriented of these books. It relies on visualization to
explain complex concepts at an intuitive level and to guide mathematical proofs. Nearly, all the
analysis in the book is geometrically motivated, and the emphasis is on rigorous, polished, and
economical arguments, which tend to reinforce the geometric intuition."

From the review by Giorgio Giorgi (Mathematical Reviews, 2012):(Full Review)

"This is another useful contribution to convex analysis and optimization by D. P. Bertsekas, a prolific author who is able to put together a rigorous treatment of the subjects and a skillful didactic presentation. .... Unlike some other books on the same subject (for example the famous book by R. T. Rockafellar ... which does not contain a single figure), the book of Bertsekas abounds in geometrical illustrations of the properties and visual treatments of the problems. ... Some results stem directly from the author's research. Some of the more standard results are not usually found in other conventional textbooks on convexity."

Among its features, the book:

develops rigorously and comprehensively the theory of convex sets
and functions, in the classical tradition of Fenchel and Rockafellar

is structured to be used conveniently either as a standalone text for a theoretically-oriented class on convex analysis and optimization, or as a theoretical supplement to either an applications/convex optimization models class or a nonlinear programming class